This work demonstrates that the four laws of classical thermodynamics apply to the statistics of symmetric observation distributions, and provides examples of how this can be exploited in uncertainty assessments. First, an expression for the partition function Z is derived. In contrast with general classical thermodynamics, however, this can be performed without the need for variational calculus, while Z also equals the number of observations N directly. Apart from the partition function Z≡N as a scaling factor, three state variables m, n, and ϵ fully statistically characterize the observation distribution, corresponding to its expectation value, degrees of freedom, and random error, respectively. Each term in the first law of thermodynamics is then shown to be a variation on δm2=δ(nϵ)2 for both canonical (constant n and ϵ) and macro-canonical (constant ϵ) observation ensembles, while micro-canonical ensembles correspond to a single observation result bin having δm2=0. This view enables the improved fitting and combining of observation distributions, capturing both measurand variability and measurement precision.
Keppens et al. (Wed,) studied this question.